Volterra operator

In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space L²[0,1] of complex-valued square-integrable functions on the interval [0,1]. On the subspace C[0,1] of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra integral equations.

Definition

The Volterra operator, V, may be defined for a function f ∈ L²[0,1] and a value t ∈ [0,1], as

V(f)(t)=\int _{0}^{t}f(s)\,ds.

Properties

V is a bounded linear operator between Hilbert spaces, with Hermitian adjoint

V^{*}(f)(t)=\int _{t}^{1}f(s)\,ds.

V is a Hilbert–Schmidt operator, hence in particular is compact.[1]
V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = {0}.[1]
V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent.
The operator norm of V is exactly ||V|| = ²⁄_π.[1]

References

"Spectrum of Indefinite Integral Operators". Stack Exchange. May 30, 2012.

Further reading

Gohberg, Israel; Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space. Providence: American Mathematical Society. ISBN 0-8218-3627-7.

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